In the early 1990s, I sat down to watch a movie that my company had just missed out on acquiring. Now, I know what my reputation is - I read the papers. I'm the supposed tough guy of the movie business. So I as sat watching this film, drenched in tears, I thought: uh-oh, if anyone sees me like this, it's not going to be good for my image. But on the other hand, who cares if this movie can affect me so deeply? And if I ever get the chance to work with this brilliant film-maker, I'm going to grab it.

Thirteen years later, I'm sitting in a dubbing theatre in London, watching the same writer-director's latest movie, with the same result - tears streaming down my face as the film comes to an end. Only now, I'm not only deeply moved, but incredibly proud, because I've done what I set out to do: get into business with this man and finance his work.

The film-maker is Anthony Minghella, whose debut, Truly, Madly, Deeply had affected me so much, and with whom I've now had the pleasure and privilege of working on three movies: The English Patient, The Talented Mr Ripley and, currently, Cold Mountain. All of them, in my book, are great movies which, in conventional industry terms, were seen as high-risk ventures. The English Patient was a hugely ambitious step up from Anthony's first two, low-budget films - in fact the original backer pulled out on the eve of production. Ripley was seen as a difficult, almost non-mainstream project because of its ambiguous central character and dark themes. But perhaps the most risky of the three was Cold Mountain, at $80m Anthony's most expensive film to date. Developed with another studio, MGM, and offered up to me as a co-production, this was the project on which, frankly, I made the greatest bet of my career.

In a repeat run of The English Patient scenario, Anthony, and in this case Miramax alongside him, found ourselves in a difficult situation when MGM had second thoughts about an admittedly high-risk project and withdrew as our partner at the 11th hour.

I immediately began searching for a studio to replace them, but none felt able to support the movie, feeling it wasn't commercial enough. And believe me, I went hat in hand to all of them. Frankly, when you get across-the-board rejection like that, it shakes you to the core, and those are the moments when you question your own judgment.

And I was torn. From a purely hard-nosed business perspective, it was a huge gamble to take on the total responsibility for such a large-scale, expensive project that tackled such weighty themes and ideas. This wasn't some comic-book franchise property that looked like a commercial home run.

But it's at times like this that you have to listen to your own inner voice, and remind yourself of how great projects sometimes are in danger of not getting made. Remember, three of Miramax's greatest successes - Pulp Fiction, Shakespeare in Love and The English Patient - were all projects dropped by other studios. Sometimes people abandon great projects that they once loved.

This is also a business where increasingly, people are reluctant to take risks. Miramax makes a whole range of films. We'll make big, commercial movies, to keep the lights on in the office; but we've never lost sight of our roots, and remain committed to making smaller movies, like Stephen Frear's tough-minded immigrant drama Dirty Pretty Things, which we did with Celador and the BBC. Or The Station Agent and The Barbarian Invasions. Or the Portuguese-language City of God, which we financed after loving the script. (I read it in English - my staff neglected to tell me the film would be in Portuguese. I said go ahead regardless.)

Cold Mountain was, unquestionably, one of the bigger risks. But artistic integrity has a way of leading us all in the right direction. We all loved the project, and I knew Anthony was already in Romania planning the movie. I just had to say the magic word. So I placed a call to Anthony, and told him - go.

If I had to identify one key thing about the project that spoke to me personally, it would be the theme of family that the film explores so movingly. One of the most important people in my life was my grandmother, who came to America with her brother from Poland right after the first world war. As children, when my brother Bob and I were anxious to avoid doing our homework, we'd fly round to her house. She would tell us stories that moved me beyond words: of the anti-semitism she faced as a young girl, and how that anti-semitism led to a riot in Poland that led to the death of her younger sister.

The courage and strength she found after going through these terrible events, and the courage and strength she drew from her surviving family, made an enormous impression on me. So when I read, in Charles Frazier's original novel and Anthony's adaptation, the tremendously powerful evocation of ordinary people surviving against impossible odds, of the most decent part of human beings rising and asserting itself above the darkest actions of which people are capable, I was moved beyond words.

So I committed to the film - more with my heart than my head, if truth be told - and I never looked back.

Given appalling weather conditions, it was an immensely tough shoot in Romania and the US over a six-month period. Anthony's insistence on shooting all four seasons was something of an issue - we kept saying CGI, CGI, CGI. Anthony, who wanted as much realism as possible, would look at us as if we were alien visitors. The suggestion was quietly dropped.

How do I feel in the end about our decision to make Cold Mountain? Only a year or so before, people had questioned the wisdom of Miramax backing another great film-maker with another epic vision - in that case, Marty Scorsese and Gangs of New York. But with a worldwide gross of over $200m, a projected revenue stream of $100m from worldwide video and DVD, 10 Oscar nominations, a Golden Globe win for Marty as best director and a Bafta for Daniel Day-Lewis, that ultimately looked like a pretty smart decision. It's too early to predict the ultimate outcome for Cold Mountain, but the film has opened to strong box office, has been embraced by numerous reviewers, and has already made its mark in terms of award nominations.

But beyond all this, for me, it will always be a film that resonates in a more profound way. Every day, we read the cold statistics of war - the loss of soldiers from Britain, from the US, and in other conflicts around the world. We read about soldiers shot down in a helicopter. For most of us, it's just another name in a newspaper. But behind every one is a family who have suffered this devastating loss, and who are going to have to find the strength and courage to go on.

I truly feel, in Cold Mountain's story of a man who has been to hell and back in wartime, and of the strength and courage found by those who survive such turbulent and momentous events, that this film gives us a powerful glimpse into that heart of darkness - and the light that awaits the survivors when they emerge.